“Dead” Exciton Layer and Exciton Anisotropy of Bulk MoS2 Extracted from Optical Measurements

Excitons (electron–hole pairs bound by the Coulomb potential) play an important role in optical and electronic properties of layered materials. They can be used to modulate light with high frequencies due to the optical Pauli blocking. The properties of excitons in 2D materials are extremely anisotropic. However, due to nanometre sizes of excitons and their short life times, reliable tools to study this anisotropy are lacking. Here, we show how direct optical reflection measurements can be used to evaluate anisotropy of excitons in transition metal dichalcogenides MoS2. Using focused beam spectroscopic ellipsometry, we have measured the polarized optical reflection of bulk MoS2 for two crystal orientations: c-axis being perpendicular to the surface from which reflection is measured and c-axis being parallel to the surface from which reflection is measured. We found that for the parallel configuration the optical reflection near excitonic transitions is strongly affected by the presence of the exciton “dead” layer such that the excitonic reflection peaks become the excitonic dips due to light interference. At the same time, the optical reflection for the perpendicular orientation is not significantly altered by the exciton “dead” layer due to large anisotropy of exciton properties. Performing simultaneous Fresnel fitting for both geometries, we were able to evaluate exciton anisotropy in layered materials from simple optical measurements.

were spatially isolated from the bulk material were chosen for further device fabrication. The optical properties of the samples were measured when light was reflected from the top surface (with c-axis being perpendicular to the surface from which reflection was measured) and from the polished edge (with c-axis being parallel to the surface from which reflection was measured). The edges of as transferred vertically aligned MoS2 flakes were rough ( Figure S1a) which could affect the results of optical measurements due to scattering. To avoid the impact of roughness, edge segments (which were sufficiently large for optical measurements) were polished using Ga focused ion beam (FIB) microscope (a hybrid FIB-SEM system, Carl Zeiss Crossbeam 540). This was done using 30 kV Ga ion beam with current decreasing from 80-100mA in 3-4 steps down to 1.5nA. On two occasions (samples W1&W2, see below) the final cross-section polishing has been subsequently completed using 5kV beam at 1.5nA.
The SEM images of the measured surfaces were obtained using two different detectors: in-lens and secondary electron detector are shown in Figures 1d and S1. Three samples have been fabricated. In the first sample, W1, a rectangular slot of 150 µm length and 50 µm width to the depth of 60 µm has been etched away using 100nA mill. Then the central area of 100x60 µm 2 was polished with decreasing currents up to 1.5nA. In the second sample, W2, a trapezoidal slot of 540 µm length and 60 µm width to the depth of 70 µm has been etched away using 100nA mill. Then the central area of 230x70 µm 2 was polished with decreasing currents up to 1.5nA. In the last sample, W3, a trapezoidal slot of ~1500 µm length and (100-150) µm width to the depth of 250 µm has been etched away using 80-100nA mill. As the system permits to etch only up to 600 µm in one direction, this has been split in 3 overlapping parts. Then the central area of 280x250 µm 2 was polished with decreasing currents down to 15 nA at 30kV. All samples showed the same optical features described in the main text. Figure S1 show in situ images obtained by SEM at =54 tilt used for milling. Note that for the tilted sample the y-direction doesn't correspond to the scale bar and should be recalculated by sin() and cos() for the cross-section and sample surface, respectively. In particular, the thickness of the sample is 1/sin54 ≈ 1.236 bigger than obtained using the scale bar. SEM images of the ac-plane from the freshly cleaved crystal after ion polishing reveals a locally smooth surface, which ensures that our optical measurement results are reliable (with illuminated area being 100200 μm 2 ), see Figs. 1e,f and

Measurement methods.
As explained in the main text, the optical properties of samples were measured in two geometries: Geometry 1 (G1), where reflection from the top surface of the transferred flakes was measured (with c-axis being perpendicular to the surface from which reflection was measured) and Geometry 2 (G2), where the reflection from the polished edge of the flake was measured (with c-axis being parallel to the surface from which reflection was measured). For G1 at normal angle of incidence, the wavevector of the light k is parallel to the unit vector c directed along the c-axis (k || c while E ⊥ c, where E is the electric field of the light wave). For G2 at normal angle of incidence k ⊥ c while we have two different orientations of the electric field E ⊥ c and E || c. It is well known that layered MoS2 crystal is built up of van der Waals bonded by S-Mo-S units. Each of these stable units consisting of two hexagonal planes of S atoms sandwiching a hexagonal plane of Mo coordinated through ionic-covalent inter-actions with each other in a trigonal prismatic arrangement 2,3 as schematically shown in Figure   1a,d. The anisotropy in the excitonic behaviours is attributed to crystal anisotropy and can be studied via the polarization dependent reflectance measurements. We found strong dependence of excitonic The edge measurements could be affected by the sample damage due to its polishing. To confirm that this is not the case and the main optical features observed in G2 geometry are not connected to polishing, we also measured reflection spectra from an unpolished flat edge of a suitable MoS2 flake.
The result of these measurements is shown in Fig The absence of excitonic features for 0-polarization as well as the change of the peaks to dips for 90 polarization cannot be simply described by the Tauc-Lorentz oscillator model 5,6 . This requires strong anisotropy of exciton properties as well as the introduction of the "dead" exciton layer suggested by Hopfield and Thomas 7 . Our experiments on non-polished-virgin, high quality polished and LiOH treated samples indicate that while surface damage and scattering may play some role in explaining optical feature near exciton transitions, the "dead" exciton layer is the most straightforward explanation of our results. Indeed, adding "dead" exciton layers of different thicknesses for the two geometries (G1 and G2) in Fresnel modelling and performing simultaneous fitting of the measured reflections in the two geometries (G1 and G2)which can be done using Wvase softwarewe can extract the thickness of the "dead" layers. This fit is shown in Figure 3 e,f and it yields the excitonfree "dead" layer thickness in G1 as 1nm and exciton free-layer "dead" layer thickness in G2 as 16nm.
The corresponding in-plane na * =na+ika and out-of-plane nc * =nc+ikc optical constants extracted form the modelling are presented in Figure S3. Note that the thick MoS2 characterized by the exciton-free "dead" layer exhibits transparent behaviour along the c-axis, even at ultraviolet and visible wavelengths (usually in region of strong interband transition for semiconductors).
Raman spectroscopy. Polarised Raman spectroscopy can be used to determine the orientation of the exfoliated anisotropic MoS2 crystals. Raman measurements were performed using a confocal scanning Raman microscope, Renishaw. All measurements were carried out using linearly polarized excitation (i.e., perpendicular and parallel polarized) with wavelength 514 nm, 1800 lines/mm diffraction grating, and ×100 objective (NA = 0.90). The laser spot size was approximately 1 μm in diameter. The laser power was less than 0.5-0.7 mW for which no physical damage or oxidation was expected to occur in the studied MoS2 flakes. Bulk molybdenum disulphide MoS2 belongs to the class of transition metal dichalcogenides (TMDs) that crystallize in the characteristic 2H polytype. The corresponding Bravais lattice is hexagonal and the space group of the crystal is D6h 4 , where the repeat unit in the c direction contains two layers and the S atoms in one layer are directly above the Mo. The unit cell is characterized by the lattice parameters a (in-plane lattice constant) and c (out-of-plane lattice constant) 8 . According to the group theory, for a perfect MoS2, the normal vibration modes at the centre of the Brillouin zone can be expressed as 9 (MoS2)=A2u(IR) + E1u(IR) + A1g(R) + 2E2g(R) + E1g(R).
In the above expression A1g, E1g, and E2g are the Raman active modes while A2u and E1u modes are infrared active. The four principal frequencies of Raman spectra for horizontal stack of MoS2 sheets are 287, 384, 408 and 447 cm -1 for both polarisations of excited light (see Figure S4a (Figure 2a,b). In accordance to Ataca et al. 10 , the weak absorption in the 1150−1000 cm −1 range observed in the IR spectra ( Figure S5b) for k ⊥ c geometry is mostly due to sulfate groups adsorbed on defect sites. This indicates that virgin MoS2 contains some adsorbed impurities, presumably formed by oxidation of the surface and likely occurring at defect sites of edges.
Thus it was shown that strong anisotropy persists up to phonon modes of the mid-IR region in MoS2 9 . The observed additional features at 1900-2000 and 3400 cm -1 probably correspond to the C=C and O-H bonds present at the surface of MoS2 sheets.

Fresnel modelling of samples.
Wvase32 software of J. A. Woollam Co., Inc., was used for modelling. The software employs Fresnel theory in order to calculate the reflection/transmission coefficients from layered samples with given optical constants. It also allows one to restore optical constants of an unknown layer placed on a given substrate from experimental measurements of ellipsometric parameters. For isotropic samples, measurements of ellipsometric parameters  and  at one angle of incidence is normally enough to extract the values of n and k of an unknown layer. For anisotropic samples, variable angle ellipsometry is used to extract optical constants in which ellipsometric reflections of an unknown sample are simultaneously fitted for several different angles of incidence. Figure 2d shows the fitting of the  spectra (the  spectra were fitted simultaneously) of thick MoS2 sheets in G1 geometry at several different angles of incidence with the help of Wvase32 software. In this modelling, MoS2 was assumed to be bi-anisotropic with different optical constants for in-plane and out-of-plane direction (MoS2 was also assumed to be semi-infinite due to large absorption). One can notice an excellent agreement of the fitted data with the experimental data. The in-plane constants are shown in Fig. 2e. Using these constants and the Wvase32 software, we calculated the reflectance Fig. 3d where we obtained peaks in the modelling data and dips in the measured data.
To address this disagreement, we have simultaneously modelled the measured ellipsometric spectra of  and  in G1 and G2 geometries acquired under several angles of incidence by adding an additional "dead layer" with different thicknesses for G1 geometry and G2 geometry (which Wvase32 software is capable of). The thickness of the "dead layers" in G1 and G2 geometries was a fitting parameter (yielding ~1nm thickness for G1 geometry and ~10nm for G2 geometry) while the optical constants of the "dead layer" were chosen to be the bi-anisotropic optical constants of MoS2 shown in Fig S3 with excitonic peaks being removed. Figure 3e and f shows the results of the modelling.